Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

When approximating the area beneath the curve $f(x)=x^2+1$ on the interval $[0,3]$ using a left Riemann sum with 4 rectangles, I calculated that the height of each rectangle, in order from left to right, would be

$f(0)$, $f(\frac{3}{4})$, $f(\frac{3}{2})$, $f(\frac{9}{4})$

and therefore, the area would be

$A=\frac{3}{4}\cdot f(0)+\frac{3}{4}\cdot f(\frac{3}{4})+\frac{3}{4}\cdot f(\frac{3}{2})+\frac{3}{4}\cdot f(\frac{9}{4})$ (I'll call this equation 1)

Now I know the next step is

$A=\frac{3}{4}\cdot1+\frac{3}{4}\cdot \frac{25}{16}+\frac{3}{4}\cdot \frac{13}{4}+\frac{3}{4}\cdot\frac{97}{16}=\frac{285}{32}\approx8.906$ (I'll call this equation 2)

But it's this step that confuses me.

How does one know for example that $f(\frac{3}{4})$ from equation 1 goes to $\frac{25}{16}$ in equation 2 and $f(\frac{9}{4})$ from equation 1 goes to $\frac{97}{16}$ in equation 2? (The same goes for the other variables).

I apologise if this is written in a confusing way.

share|improve this question

2 Answers 2

up vote 4 down vote accepted

Well $f(x) = x^2+1$

$$f\left(\frac 3 4\right) = \left(\frac 3 4\right)^2+1 = \frac 9 {16} + \frac {16}{16} = \frac {25} {16}$$ and $$f\left(\frac 9 4\right) = \left(\frac 9 4\right)^2+1 = \frac {81} {16} + \frac {16}{16} = \frac {97} {16}$$

share|improve this answer
1  
Ah my goodness of course, how did I not see this, I feel so stupid! –  Olly Price Aug 15 '12 at 14:36
    
You are applying the function to the leftmost side of the rectangle to get the left sum. –  Mark Bennet Aug 15 '12 at 14:37

The values such as $\frac{25}{16}$ and $\frac{97}{16}$ were obtained by evaluating the original function $f(x) = x^2 +1$. Namely:

$f(0) = 0^2+1 = 1$

$f(\frac{3}{4}) = (\frac{3}{4})^2 + 1 = \frac{25}{16}$

$f(\frac{3}{2}) = (\frac{3}{2})^2 + 1 = \frac{13}{4}$

$f(\frac{9}{4}) = (\frac{9}{4})^2 + 1 = \frac{97}{16}$

Each value is used to estimate the height of the curve for a certain segment of the interval $[0,3]$.

For example, the value $f(0)=1$ is an estimate for the height of the curve for the first segment, $[0,\frac{3}{4}]$. So the area of the first rectangle is $\frac{3}{4} . 1 = \frac{3}{4}$

A similar calculation is done for all 4 rectangles, and then sum them up to get the approximate total area under the curve.

If you increase the number of rectangles in your approximation, you will generally get a better estimate for the area under the curve.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.